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Please help me with this. I need to find a non-trivial function $g(x)$ which satisfy the following functional equation $$\tan(g(x))+g(x)+g(x)\tan^2(g(x))=0$$

Darek
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  • This is not a functional equation. – Ivan Neretin Feb 01 '19 at 11:24
  • I am afraid you are wrong. This is a functional equation. – Darek Feb 01 '19 at 13:49
  • OK, this is a functional equation in name only. Imagine a simple algebraic equation, like $x^3-3x^2-3x+1=0$. It is definitely not a functional equation, for there is no unknown function of any sort, is it? Now suppose that we rewrite this equation and substitute every single instance of $x$ with $f(t)$. Does this make it a functional equation? Technically yes, but... – Ivan Neretin Feb 01 '19 at 13:58

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There is no non-trivial real (I assume you meant real as you didn't specify anything else) solution. We can rewrite your equation (using the usual trigonometric identities) as

$$g(x) = - \frac{\tan(g(x))}{1+\tan^2(g(x))} = -\frac{1}{2} \sin(2 g(x))$$

If there is some $x_0 \in \mathbb R$ such that $a := g(x_0) \neq 0$ then we have

$$a = - \frac{1}{2} \sin(2 a)$$

which is equivalent to $-2a = \sin(2a)$ which has the only real solution $a=0$. Therefore $g(x_0)=0$ and therefore $g(x) = 0$ for all $x$.

flawr
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